June  2018, 12(3): 667-676. doi: 10.3934/ipi.2018028

EIT in a layered anisotropic medium

1. 

Dipartimento di Matematica e Geoscienze, Università di Trieste, Via Valerio 12/1 -34127, Trieste, Italy

2. 

Departments of Computational and Applied Mathematics, Earth Science, Rice University, Houston, Texas, USA

3. 

Department of Mathematics and Statistics, Health Research Institute (HRI), University of Limerick, Castletroy, Limerick, V94 T9PX, Ireland

Received  August 2017 Revised  December 2017 Published  March 2018

We consider the inverse problem in geophysics of imaging the subsurface of the Earth in cases where a region below the surface is known to be formed by strata of different materials and the depths and thicknesses of the strata and the (possibly anisotropic) conductivity of each of them need to be identified simultaneously. This problem is treated as a special case of the inverse problem of determining a family of nested inclusions in a medium $Ω\subset\mathbb{R}^n$, $n ≥ 3$.

Citation: Giovanni Alessandrini, Maarten V. de Hoop, Romina Gaburro, Eva Sincich. EIT in a layered anisotropic medium. Inverse Problems and Imaging, 2018, 12 (3) : 667-676. doi: 10.3934/ipi.2018028
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Pure and Applied Mathematics (Amsterdam), 140, 2nd edition, Elsevier/Academic Press, Amsterdam, 2003.

[2]

G. Alessandrini, Singular solutions of elliptic equations and the determination of conductivity by boundary measurements, J. Differential Equations, 84 (1990), 252-272.  doi: 10.1016/0022-0396(90)90078-4.

[3]

G. AlessandriniM. V. de Hoop and R. Gaburro, Uniqueness for the electrostatic inverse boundary value problem with piecewise constant anisotropic conductivities, Inverse Problems, 33 (2017), 125013. 

[4]

G. Alessandrini and R. Gaburro, Determining conductivity with special anisotropy by boundary measurements, SIAM J. Math. Anal., 33 (2001), 153-171.  doi: 10.1137/S0036141000369563.

[5]

G. Alessandrini and R. Gaburro, The local Calderón problem and the determination at the boundary of the conductivity, Comm. Partial Differential Equations, 34 (2009), 918-936.  doi: 10.1080/03605300903017397.

[6]

K. AstalaM. Lassas and L. Päivärinta, Calderón inverse problem for anisotropic conductivity in the plane, Comm. Partial Differential Equations, 30 (2005), 207-224.  doi: 10.1081/PDE-200044485.

[7]

M. I. Belishev, The Calderón problem for two-dimensional manifolds by the BC-Method, SIAM J. Math. Anal., 35 (2003), 172-182.  doi: 10.1137/S0036141002413919.

[8]

E. Bierstone and P. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math., 67 (1988), 5-42. 

[9]

A. P. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), 65–73, Soc. Brasil. Mat., Rio de Janeiro, 1980. Reprinted in: Comput. Appl. Math., 25 (2006), 133–138.

[10]

C. I. Cârstea, N. Honda and G. Nakamura, Uniqueness in the inverse boundary value problem for piecewise homogeneous anisotropic elasticity, preprint, arXiv: 1611.03930.

[11]

Ellis R. G. and D. W. Oldenburg, The pole-pole 3-D DC-resistivity inverse problem: A conjugate gradient approach, Geophysical Journal International, 119 (1994), 187-194. 

[12]

C. G. Farquharson, Constructing piecewise-constant models in multidimensional minimum-structure inversions, Geophysics, 73 (2007), K1-K9.  doi: 10.1190/1.2816650.

[13]

R. Gaburro and W. R. B. Lionheart, Recovering Riemannian metrics in monotone families from boundary data, Inverse Problems, 25 (2009), 045004, 14 pp. 

[14]

R. Gaburro and E. Sincich, Lipschitz stability for the inverse conductivity problem for a conformal class of anisotropic conductivities, Inverse Problems, 31 (2015), 015008, 26pp. 

[15]

L. A. Gallardo and M. A. Meju, Joint two-dimensional DC resistivity and seismic travel time inversion with cross-gradients constraints, Journal of Geophysical Research, 109 (2004), B03311.  doi: 10.1029/2003JB002716.

[16]

T. GüntherC. Rücker and K. Spitzer, Three-dimensional modelling and inversion of dc resistivity data incorporating topography Ⅱ. Inversion, Geophysical Journal International, 166 (2006), 506-517. 

[17]

E. Haber and D. Oldenburg, Joint inversion: A structural approach, Inverse Problems, 13 (1997), 63-77.  doi: 10.1088/0266-5611/13/1/006.

[18]

M. Ikehata, Identification of the curve of discontinuity of the determinant of the anisotropic conductivity, J. Inverse Ill-Posed Probl., 8 (2000), 273-285. 

[19]

R. Kohn and M. Vogelius, Identification of an unknown conductivity by means of measurements at the boundary, SIAM-AMS Proc., 14 (1984), 113-123. 

[20]

M. Lassas and G. Uhlmann, On determining a Riemannian manifold from the Dirichlet-to-Neumann map, Ann. Sci. École Norm. Sup., 34 (2001), 771-787.  doi: 10.1016/S0012-9593(01)01076-X.

[21]

M. LassasG. Uhlmann and M. Taylor, The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary, Comm. Anal. Geom., 11 (2003), 207-221.  doi: 10.4310/CAG.2003.v11.n2.a2.

[22]

J. M. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42 (1989), 1097-1112.  doi: 10.1002/cpa.3160420804.

[23]

W. R. B. Lionheart, Conformal uniqueness results in anisotropic electrical impedance imaging, Inverse Problems, 13 (1997), 125-134.  doi: 10.1088/0266-5611/13/1/010.

[24]

M. H. LokeI. Acworth and T. Dahlin, A comparison of smooth and blocky inversion methods in 2D electrical imaging surveys, Exploration Geophysics, 34 (2003), 182-187.  doi: 10.1071/EG03182.

[25]

A. Malinverno and C. Torres-Verdín, Bayesian inversion of DC electrical measurements with uncertainties for reservoir monitoring, Inverse Problems, 16 (2000), 1343-1356.  doi: 10.1088/0266-5611/16/5/313.

[26]

A. Nachman, Global uniqueness for a two dimensional inverse boundary value problem, Ann. Math., 143 (1995), 71-96.  doi: 10.2307/2118653.

[27]

A. Paré and Y. Li, Improved imaging of sharp boundaries in DC resistivity, SEG International Exposition and Annual Meeting (SEG-2017-17739005), (2017), 24-29. 

[28]

J. Sylvester, An anisotropic inverse boundary value problem, Comm. Pure. Appl. Math., 43 (1990), 201-232.  doi: 10.1002/cpa.3160430203.

[29]

G. Uhlmann, Electrical impedance tomography and Calder´on's problem (topical review), Inverse Problems, 25 (2009), 123011, 39pp. 

[30]

J. ZhangR. L. Mackie and T. R. Madden, 3-D resistivity forward modeling and inversion using conjugate gradients, SEG Technical Program Expanded Abstracts, (1994), 377-380.  doi: 10.1190/1.1932101.

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Pure and Applied Mathematics (Amsterdam), 140, 2nd edition, Elsevier/Academic Press, Amsterdam, 2003.

[2]

G. Alessandrini, Singular solutions of elliptic equations and the determination of conductivity by boundary measurements, J. Differential Equations, 84 (1990), 252-272.  doi: 10.1016/0022-0396(90)90078-4.

[3]

G. AlessandriniM. V. de Hoop and R. Gaburro, Uniqueness for the electrostatic inverse boundary value problem with piecewise constant anisotropic conductivities, Inverse Problems, 33 (2017), 125013. 

[4]

G. Alessandrini and R. Gaburro, Determining conductivity with special anisotropy by boundary measurements, SIAM J. Math. Anal., 33 (2001), 153-171.  doi: 10.1137/S0036141000369563.

[5]

G. Alessandrini and R. Gaburro, The local Calderón problem and the determination at the boundary of the conductivity, Comm. Partial Differential Equations, 34 (2009), 918-936.  doi: 10.1080/03605300903017397.

[6]

K. AstalaM. Lassas and L. Päivärinta, Calderón inverse problem for anisotropic conductivity in the plane, Comm. Partial Differential Equations, 30 (2005), 207-224.  doi: 10.1081/PDE-200044485.

[7]

M. I. Belishev, The Calderón problem for two-dimensional manifolds by the BC-Method, SIAM J. Math. Anal., 35 (2003), 172-182.  doi: 10.1137/S0036141002413919.

[8]

E. Bierstone and P. Milman, Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math., 67 (1988), 5-42. 

[9]

A. P. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), 65–73, Soc. Brasil. Mat., Rio de Janeiro, 1980. Reprinted in: Comput. Appl. Math., 25 (2006), 133–138.

[10]

C. I. Cârstea, N. Honda and G. Nakamura, Uniqueness in the inverse boundary value problem for piecewise homogeneous anisotropic elasticity, preprint, arXiv: 1611.03930.

[11]

Ellis R. G. and D. W. Oldenburg, The pole-pole 3-D DC-resistivity inverse problem: A conjugate gradient approach, Geophysical Journal International, 119 (1994), 187-194. 

[12]

C. G. Farquharson, Constructing piecewise-constant models in multidimensional minimum-structure inversions, Geophysics, 73 (2007), K1-K9.  doi: 10.1190/1.2816650.

[13]

R. Gaburro and W. R. B. Lionheart, Recovering Riemannian metrics in monotone families from boundary data, Inverse Problems, 25 (2009), 045004, 14 pp. 

[14]

R. Gaburro and E. Sincich, Lipschitz stability for the inverse conductivity problem for a conformal class of anisotropic conductivities, Inverse Problems, 31 (2015), 015008, 26pp. 

[15]

L. A. Gallardo and M. A. Meju, Joint two-dimensional DC resistivity and seismic travel time inversion with cross-gradients constraints, Journal of Geophysical Research, 109 (2004), B03311.  doi: 10.1029/2003JB002716.

[16]

T. GüntherC. Rücker and K. Spitzer, Three-dimensional modelling and inversion of dc resistivity data incorporating topography Ⅱ. Inversion, Geophysical Journal International, 166 (2006), 506-517. 

[17]

E. Haber and D. Oldenburg, Joint inversion: A structural approach, Inverse Problems, 13 (1997), 63-77.  doi: 10.1088/0266-5611/13/1/006.

[18]

M. Ikehata, Identification of the curve of discontinuity of the determinant of the anisotropic conductivity, J. Inverse Ill-Posed Probl., 8 (2000), 273-285. 

[19]

R. Kohn and M. Vogelius, Identification of an unknown conductivity by means of measurements at the boundary, SIAM-AMS Proc., 14 (1984), 113-123. 

[20]

M. Lassas and G. Uhlmann, On determining a Riemannian manifold from the Dirichlet-to-Neumann map, Ann. Sci. École Norm. Sup., 34 (2001), 771-787.  doi: 10.1016/S0012-9593(01)01076-X.

[21]

M. LassasG. Uhlmann and M. Taylor, The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary, Comm. Anal. Geom., 11 (2003), 207-221.  doi: 10.4310/CAG.2003.v11.n2.a2.

[22]

J. M. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42 (1989), 1097-1112.  doi: 10.1002/cpa.3160420804.

[23]

W. R. B. Lionheart, Conformal uniqueness results in anisotropic electrical impedance imaging, Inverse Problems, 13 (1997), 125-134.  doi: 10.1088/0266-5611/13/1/010.

[24]

M. H. LokeI. Acworth and T. Dahlin, A comparison of smooth and blocky inversion methods in 2D electrical imaging surveys, Exploration Geophysics, 34 (2003), 182-187.  doi: 10.1071/EG03182.

[25]

A. Malinverno and C. Torres-Verdín, Bayesian inversion of DC electrical measurements with uncertainties for reservoir monitoring, Inverse Problems, 16 (2000), 1343-1356.  doi: 10.1088/0266-5611/16/5/313.

[26]

A. Nachman, Global uniqueness for a two dimensional inverse boundary value problem, Ann. Math., 143 (1995), 71-96.  doi: 10.2307/2118653.

[27]

A. Paré and Y. Li, Improved imaging of sharp boundaries in DC resistivity, SEG International Exposition and Annual Meeting (SEG-2017-17739005), (2017), 24-29. 

[28]

J. Sylvester, An anisotropic inverse boundary value problem, Comm. Pure. Appl. Math., 43 (1990), 201-232.  doi: 10.1002/cpa.3160430203.

[29]

G. Uhlmann, Electrical impedance tomography and Calder´on's problem (topical review), Inverse Problems, 25 (2009), 123011, 39pp. 

[30]

J. ZhangR. L. Mackie and T. R. Madden, 3-D resistivity forward modeling and inversion using conjugate gradients, SEG Technical Program Expanded Abstracts, (1994), 377-380.  doi: 10.1190/1.1932101.

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